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Metamath Proof Explorer

Theorem wuncnv

Description: A weak universe is closed under the converse operator. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypotheses wun0.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
wunop.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
Assertion wuncnv ( 𝜑 → ◡ 𝐴 ∈ 𝑈 )

Proof

Step Hyp Ref Expression
1 wun0.1 ⊢ ( 𝜑 → 𝑈 ∈ WUni )
2 wunop.2 ⊢ ( 𝜑 → 𝐴 ∈ 𝑈 )
3 1 2 wunrn ⊢ ( 𝜑 → ran 𝐴 ∈ 𝑈 )
4 1 2 wundm ⊢ ( 𝜑 → dom 𝐴 ∈ 𝑈 )
5 1 3 4 wunxp ⊢ ( 𝜑 → ( ran 𝐴 × dom 𝐴 ) ∈ 𝑈 )
6 cnvssrndm ⊢ ◡ 𝐴 ⊆ ( ran 𝐴 × dom 𝐴 )
7 6 a1i ⊢ ( 𝜑 → ◡ 𝐴 ⊆ ( ran 𝐴 × dom 𝐴 ) )
8 1 5 7 wunss ⊢ ( 𝜑 → ◡ 𝐴 ∈ 𝑈 )